Mathematics

Friedrich Wilhelm Nietzsche is probably the most controversial philosopher who has ever lived in the Western tradition. We will examine his ideas chronologically, from his early work on Greek tragedy to his excoriating criticisms...

This map is a covering map by the same argument from Theorem 2.p on its own takes each unit interval in ℝ and wraps it around the unit circle. p×p takes each unit square × where n,m∈ℝ and wraps it around the torus.In reality...

The application of the first generation blockchain network was electronic cash. Since then, enhancements have been made to the protocol. The next generation blockchain technology is a single programmable blockchain network which can be...

Definition 1.1Let f:X⟶Yand f′:X⟶Ybe two continuous maps of the space X into the space Y. We say that f is homotopic to f′, written f≃f′, if there exists a continuous map F:X×I⟶Ywhere I denotes the interval , such that F(x,0)=f(x)and F(x,1)=f′(x)The map F is called a homotopy between f and f′. If f≃f′ and f′ is a constant map, we call f null-homotopic. In other words,...

Definition 2.1 Let X be a space and let x0∈X be an element in that space. We call a path in X that begins and ends at x0 a loop based at x0. The set of path homotopy classes of loops based at x0, with operation ∗ is...

The Riemann Integral is not Enough For the majority of functions you will come across on a regular basis, Riemann integration will work just fine. However, this doesn’t mean that this approach to the integral...

The Riemann Integral Riemann integration is a topic you are likely familiar with from calculus class. Here, we are going to define the Riemann integral in a way you may not be used to. This...

Probability Theory Probability Space To start our discussion of probability theory, we first need to define the concept of probability space. For a simple definition, the probability space of some probabilistic experiment is the set of...